Re: ? dual spaces

[Cheng Cosine <acosine@xxxxxxxxxx>]
Or you're trying to emphasize that when A is linear, a subspace U of
A's domain and of say 3 dimension must have a subspace V of A's codomain
to be 3 dimensional as well.

If A is a linear operator defined on a finite dimensional
linear space U, the following is true:

dim A(U) <= dim U .

This is easy to see, because given a basis {u1, ..., uN} for
U, its image {A(u1), ..., A(uN)} spans A(U).

However, when A is nonlinear, this is not
true in general. The dimension of the subspace in codomain can be of any
number. But a sapce is always linear and has basis and dimension?

No. The image of a linear space U under a _nonlinear_
operator A is typically not a linear subspace, meaning that
the linear combination of two elements of A(U) may not be in
A(U). Therefore, you cannot in general find a basis that
spans only A(U). Also the notion of dimension (in the sense
of linear algebra) does not apply to such sets.

[...]

HTH,

--
Pouya D. Tafti
p dot d dot tafti at ieee dot org
.

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